A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.
","abstract_html":"<p>A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.</p>","abstract_has_math":false,"creators":["Hearon, Sean M"],"institution":null,"degree_name":"Master of Arts in Mathematics","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Aikin, Jeremy"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-12-01T08:00:00Z","date_published":"2016-12-01T08:00:00Z","updated_at":"2026-07-24T01:53:07Z","subjects":["Graphs","Combinatorics","Planar","Biplanar","Graph thickness","Discrete Mathematics and Combinatorics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.lib.csusb.edu/etd/427","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Aikin, Jeremy"]},{"key":"dc:creator","label":"Author","values":["Hearon, Sean M"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2016-11-10T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Arts in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Graphs","Combinatorics","Planar","Biplanar","Graph thickness","Discrete Mathematics and Combinatorics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.lib.csusb.edu/etd/427"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.
"]},{"key":"dc:title","label":"Title","values":["PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS"]}]}],"canonical_facts":{"dc:contributor":["Aikin, Jeremy"],"dc:creator":["Hearon, Sean M"],"dc:date.available":["2016-11-10T08:00:00Z"],"dc:description.abstract":["A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.
"],"dc:identifier":["https://scholarworks.lib.csusb.edu/etd/427"],"dc:subject":["Graphs","Combinatorics","Planar","Biplanar","Graph thickness","Discrete Mathematics and Combinatorics"],"dc:title":["PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Arts in Mathematics"]},"updated_at":"2026-07-24T01:53:07Z"}